Continuous Functions

Continuous functions in topology are maps between spaces that preserve their structure, allowing deformations without breaks.

Definition

A function \(f: X \to Y\) between topological spaces is continuous if the preimage of every open set in \(Y\) is open in \(X\).

Examples

The function \(f(x) = x^2\) from \(\mathbb{R}\) to \(\mathbb{R}\) is continuous; preimages of open intervals are open intervals.

Properties

Continuous functions preserve connectedness and compactness, key topological invariants.